Calculus Examples

$y = \ln (x^{6} (x^{9} - x + 5))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (x^{6} (x^{9} - x + 5)))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{6} (x^{9} - x + 5)$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x^{6} (x^{9} - x + 5)$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{6} (x^{9} - x + 5)]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [x^{6} (x^{9} - x + 5)]$

Step 3.1.3

Replace all occurrences of $u$ with $x^{6} (x^{9} - x + 5)$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} \frac{d}{d x} [x^{6} (x^{9} - x + 5)]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{6}$ and $g (x) = x^{9} - x + 5$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} \frac{d}{d x} [x^{9} - x + 5] + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $x^{9} - x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{9}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (\frac{d}{d x} [x^{9}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5]) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} + \frac{d}{d x} [- x] + \frac{d}{d x} [5]) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - \frac{d}{d x} [x] + \frac{d}{d x} [5]) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - 1 \cdot 1 + \frac{d}{d x} [5]) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.5

Multiply $- 1$ by $1$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - 1 + \frac{d}{d x} [5]) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.6

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - 1 + 0) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.7

Add $9 x^{8} - 1$ and $0$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - 1) + (x^{9} - x + 5) \frac{d}{d x} [x^{6}])$

Step 3.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - 1) + (x^{9} - x + 5) (6 x^{5}))$

Step 3.3.9

Move $6$ to the left of $x^{9} - x + 5$ .

$\frac{1}{x^{6} (x^{9} - x + 5)} (x^{6} (9 x^{8} - 1) + 6 \cdot (x^{9} - x + 5) x^{5})$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$\frac{1}{x^{6} x^{9} + x^{6} (- x) + x^{6} \cdot 5} (x^{6} (9 x^{8} - 1) + 6 (x^{9} - x + 5) x^{5})$

Step 3.4.2

Apply the distributive property.

$\frac{1}{x^{6} x^{9} + x^{6} (- x) + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 (x^{9} - x + 5) x^{5})$

Step 3.4.3

Apply the distributive property.

$\frac{1}{x^{6} x^{9} + x^{6} (- x) + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + (6 x^{9} + 6 (- x) + 6 \cdot 5) x^{5})$

Step 3.4.4

Apply the distributive property.

$\frac{1}{x^{6} x^{9} + x^{6} (- x) + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5

Combine terms.

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Step 3.4.5.1

Multiply $x^{6}$ by $x^{9}$ by adding the exponents.

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Step 3.4.5.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{6 + 9} + x^{6} (- x) + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.1.2

Add $6$ and $9$ .

$\frac{1}{x^{15} + x^{6} (- x) + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.2

Multiply $x^{6}$ by $x$ by adding the exponents.

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Step 3.4.5.2.1

Move $x$ .

$\frac{1}{x^{15} + x \cdot x^{6} \cdot - 1 + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.2.2

Multiply $x$ by $x^{6}$ .

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Step 3.4.5.2.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{x^{15} + x^{1} x^{6} \cdot - 1 + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{15} + x^{1 + 6} \cdot - 1 + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.2.3

Add $1$ and $6$ .

$\frac{1}{x^{15} + x^{7} \cdot - 1 + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.3

Move $- 1$ to the left of $x^{7}$ .

$\frac{1}{x^{15} - 1 \cdot x^{7} + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.4

Rewrite $- 1 x^{7}$ as $- x^{7}$ .

$\frac{1}{x^{15} - x^{7} + x^{6} \cdot 5} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.5

Move $5$ to the left of $x^{6}$ .

$\frac{1}{x^{15} - x^{7} + 5 \cdot x^{6}} (x^{6} (9 x^{8}) + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.6

Multiply $x^{6}$ by $x^{8}$ by adding the exponents.

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Step 3.4.5.6.1

Move $x^{8}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (x^{8} x^{6} \cdot 9 + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (x^{8 + 6} \cdot 9 + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.6.3

Add $8$ and $6$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (x^{14} \cdot 9 + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.7

Move $9$ to the left of $x^{14}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 \cdot x^{14} + x^{6} \cdot - 1 + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.8

Move $- 1$ to the left of $x^{6}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - 1 \cdot x^{6} + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.9

Rewrite $- 1 x^{6}$ as $- x^{6}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{9} x^{5} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.10

Multiply $x^{9}$ by $x^{5}$ by adding the exponents.

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Step 3.4.5.10.1

Move $x^{5}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 (x^{5} x^{9}) + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.10.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{5 + 9} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.10.3

Add $5$ and $9$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} + 6 (- x) x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.11

Multiply $- 1$ by $6$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} - 6 x \cdot x^{5} + 6 \cdot 5 x^{5})$

Step 3.4.5.12

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 3.4.5.12.1

Move $x^{5}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} - 6 (x^{5} x) + 6 \cdot 5 x^{5})$

Step 3.4.5.12.2

Multiply $x^{5}$ by $x$ .

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Step 3.4.5.12.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} - 6 (x^{5} x^{1}) + 6 \cdot 5 x^{5})$

Step 3.4.5.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} - 6 x^{5 + 1} + 6 \cdot 5 x^{5})$

Step 3.4.5.12.3

Add $5$ and $1$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} - 6 x^{6} + 6 \cdot 5 x^{5})$

Step 3.4.5.13

Multiply $6$ by $5$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (9 x^{14} - x^{6} + 6 x^{14} - 6 x^{6} + 30 x^{5})$

Step 3.4.5.14

Add $9 x^{14}$ and $6 x^{14}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (15 x^{14} - x^{6} - 6 x^{6} + 30 x^{5})$

Step 3.4.5.15

Subtract $6 x^{6}$ from $- x^{6}$ .

$\frac{1}{x^{15} - x^{7} + 5 x^{6}} (15 x^{14} - 7 x^{6} + 30 x^{5})$

Step 3.4.6

Reorder the factors of $\frac{1}{x^{15} - x^{7} + 5 x^{6}} (15 x^{14} - 7 x^{6} + 30 x^{5})$ .

$(15 x^{14} - 7 x^{6} + 30 x^{5}) \frac{1}{x^{15} - x^{7} + 5 x^{6}}$

Step 3.4.7

Factor $x^{6}$ out of $x^{15} - x^{7} + 5 x^{6}$ .

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Step 3.4.7.1

Factor $x^{6}$ out of $x^{15}$ .

$(15 x^{14} - 7 x^{6} + 30 x^{5}) \frac{1}{x^{6} x^{9} - x^{7} + 5 x^{6}}$

Step 3.4.7.2

Factor $x^{6}$ out of $- x^{7}$ .

$(15 x^{14} - 7 x^{6} + 30 x^{5}) \frac{1}{x^{6} x^{9} + x^{6} (- x) + 5 x^{6}}$

Step 3.4.7.3

Factor $x^{6}$ out of $5 x^{6}$ .

$(15 x^{14} - 7 x^{6} + 30 x^{5}) \frac{1}{x^{6} x^{9} + x^{6} (- x) + x^{6} \cdot 5}$

Step 3.4.7.4

Factor $x^{6}$ out of $x^{6} x^{9} + x^{6} (- x)$ .

$(15 x^{14} - 7 x^{6} + 30 x^{5}) \frac{1}{x^{6} (x^{9} - x) + x^{6} \cdot 5}$

Step 3.4.7.5

Factor $x^{6}$ out of $x^{6} (x^{9} - x) + x^{6} \cdot 5$ .

$(15 x^{14} - 7 x^{6} + 30 x^{5}) \frac{1}{x^{6} (x^{9} - x + 5)}$

Step 3.4.8

Multiply $15 x^{14} - 7 x^{6} + 30 x^{5}$ by $\frac{1}{x^{6} (x^{9} - x + 5)}$ .

$\frac{15 x^{14} - 7 x^{6} + 30 x^{5}}{x^{6} (x^{9} - x + 5)}$

Step 3.4.9

Factor $x^{5}$ out of $15 x^{14} - 7 x^{6} + 30 x^{5}$ .

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Step 3.4.9.1

Factor $x^{5}$ out of $15 x^{14}$ .

$\frac{x^{5} (15 x^{9}) - 7 x^{6} + 30 x^{5}}{x^{6} (x^{9} - x + 5)}$

Step 3.4.9.2

Factor $x^{5}$ out of $- 7 x^{6}$ .

$\frac{x^{5} (15 x^{9}) + x^{5} (- 7 x) + 30 x^{5}}{x^{6} (x^{9} - x + 5)}$

Step 3.4.9.3

Factor $x^{5}$ out of $30 x^{5}$ .

$\frac{x^{5} (15 x^{9}) + x^{5} (- 7 x) + x^{5} \cdot 30}{x^{6} (x^{9} - x + 5)}$

Step 3.4.9.4

Factor $x^{5}$ out of $x^{5} (15 x^{9}) + x^{5} (- 7 x)$ .

$\frac{x^{5} (15 x^{9} - 7 x) + x^{5} \cdot 30}{x^{6} (x^{9} - x + 5)}$

Step 3.4.9.5

Factor $x^{5}$ out of $x^{5} (15 x^{9} - 7 x) + x^{5} \cdot 30$ .

$\frac{x^{5} (15 x^{9} - 7 x + 30)}{x^{6} (x^{9} - x + 5)}$

Step 3.4.10

Cancel the common factors.

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Step 3.4.10.1

Factor $x^{5}$ out of $x^{6} (x^{9} - x + 5)$ .

$\frac{x^{5} (15 x^{9} - 7 x + 30)}{x^{5} (x (x^{9} - x + 5))}$

Step 3.4.10.2

Cancel the common factor.

$\frac{x^{5} (15 x^{9} - 7 x + 30)}{x^{5} (x (x^{9} - x + 5))}$

Step 3.4.10.3

Rewrite the expression.

$\frac{15 x^{9} - 7 x + 30}{x (x^{9} - x + 5)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{15 x^{9} - 7 x + 30}{x (x^{9} - x + 5)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{15 x^{9} - 7 x + 30}{x (x^{9} - x + 5)}$